Digital imaging devices such as cameras and scanners generally capture an image of a “real world” object and map this image onto a grid of pixels (also referred to as the sampled image). The sampled image is typically not a completely faithful reproduction of the real world object. For instance, considering a camera having a digital sensor made up is of a rectangular array of sensors, spatial inaccuracies introduced by the imaging system of the camera typically cause the sampled image not to be a faithful reproduction of object being imaged. A common example of such inaccuracy is referred to a “barrel distortion” which arises when the light sampled by the digital sensor of a camera is radially distorted by optical effects caused by imperfections in the lens of the camera.
Another example of an imaging system that introduces spatial distortion is a flatbed scanner. In this case the image is generally sampled by a line (or lines for colour scanners) of sensors that is moved across the platen of the scanner by a drive motor. Flatbed scanners suffer from optical distortions similar to those that affect cameras due to the presence of imaging optics. Flatbed scanners also suffer from spatial distortions due to variations in the rate at which the line of sensors is driven across the scanner platen.
Spatial distortion is also a common problem in printing systems, particularly electro-photographic printing systems. In an electro-photographic printing system, an image is formed on a piece of paper by transferring toner from a light-sensitive drum to the paper (perhaps via a transfer belt) and then by fixing the toner to the paper by applying heat. The electro-photographic drum can have characteristic errors in its operation, such as not being perfectly circular, or its axis not being correctly positioned relative to its circumference. This leads to spatial distortions that reduce the quality of the printed output.
One method of characterizing (i.e. measuring) spatial distortions introduced by imaging devices is to measure the distortions using a known test pattern (referred to as a reference test pattern or a reference calibration pattern) whose luminance is modulated in a two-dimensional manner using two sinusoidal carriers, where the phase of the modulating sinusoid varies linearly as a function of distance along two orthogonal axes (i.e. an X axis and a Y axis). This test pattern is typically generated as a pattern of dots on a medium, and the resultant chart referred to as a calibration target. By independently demodulating these sinusoids from the image of the known test pattern (referred to as the captured test pattern, or the captured calibration pattern, or the sampled image) captured by the device under test, converting the demodulated sinusoids into two complex phase functions, and unwrapping the periodic phase into continuous phase functions, the X component of any spatial distortion is manifested as an offset in the X phase function, and the Y component of any spatial distortion is manifested as an offset in the Y phase function.
While this method is typically quite effective, there are many practical considerations which affect the accuracy and the range of conditions over which it can be usefully applied. If the sinusoidal frequencies are too low, then the two carriers will potentially interfere with each other causing errors in the measured results. In addition, the phase functions are adversely affected by noise in the measured image. Furthermore, rapid changes in the spatial distortion relative to the wavelength of the sinusoidal carriers may not be resolvable. In addition, if the sinusoidal frequencies are too high, then the frequency shift introduced by the spatial distortion can cause aliasing, meaning that an accurate estimate of the spatial distortion cannot be recovered directly.